Choquet integral

<h2 id="definition">Definition</h2>
<p>The following notation is used:
</p>
<ul><li>
  
    
      
        S
      
    
    {\displaystyle S}
  
 – a set.</li>
<li>
  
    
      
        
          
            F
          
        
      
    
    {\displaystyle {\mathcal {F}}}
  
 – a collection of subsets of 
  
    
      
        S
      
    
    {\displaystyle S}
  
.</li>
<li>
  
    
      
        f
        :
        S
        →
        
          R
        
      
    
    {\displaystyle f:S\to \mathbb {R} }
  
 – a function.</li>
<li>
  
    
      
        ν
        :
        
          
            F
          
        
        →
        
          
            R
          
          
            +
          
        
      
    
    {\displaystyle \nu :{\mathcal {F}}\to \mathbb {R} ^{+}}
  
 – a monotone <a href="/facts/Set_function/Dy46B9EJ">set function</a>.</li></ul>
<p>Assume that 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is measurable with respect to 
  
    
      
        
          
            F
          
        
      
    
    {\displaystyle {\mathcal {F}}}
  
, that is
</p>

∀
        x
        ∈
        
          R
        
        :
        {
        s
        ∈
        S
        ∣
        f
        (
        s
        )
        ≥
        x
        }
        ∈
        
          
            F
          
        
      
    
    {\displaystyle \forall x\in \mathbb {R} \colon \{s\in S\mid f(s)\geq x\}\in {\mathcal {F}}}

<p>Then the Choquet integral of 
  
    
      
        f
      
    
    {\displaystyle f}
  
 with respect to 
  
    
      
        ν
      
    
    {\displaystyle \nu }
  
 is defined by:
</p>

(
        C
        )
        ∫
        f
        d
        ν
        :=
        
          ∫
          
            −
            ∞
          
          
            0
          
        
        (
        ν
        (
        {
        s
        
          |
        
        f
        (
        s
        )
        ≥
        x
        }
        )
        −
        ν
        (
        S
        )
        )
        
        d
        x
        +
        
          ∫
          
            0
          
          
            ∞
          
        
        ν
        (
        {
        s
        
          |
        
        f
        (
        s
        )
        ≥
        x
        }
        )
        
        d
        x
      
    
    {\displaystyle (C)\int fd\nu :=\int _{-\infty }^{0}(\nu (\{s|f(s)\geq x\})-\nu (S))\,dx+\int _{0}^{\infty }\nu (\{s|f(s)\geq x\})\,dx}

<p>where the integrals on the right-hand side are the usual <a href="/facts/Riemann_integral/vWGrPhVh">Riemann integral</a> (the integrands are integrable because they are monotone in 
  
    
      
        x
      
    
    {\displaystyle x}
  
).
</p>
<h2 id="properties">Properties</h2>
<p>In general the Choquet integral does not satisfy additivity. More specifically, if 
  
    
      
        ν
      
    
    {\displaystyle \nu }
  
 is not a probability measure, it may hold that
</p>

∫
        f
        
        d
        ν
        +
        ∫
        g
        
        d
        ν
        ≠
        ∫
        (
        f
        +
        g
        )
        
        d
        ν
        .
      
    
    {\displaystyle \int f\,d\nu +\int g\,d\nu \neq \int (f+g)\,d\nu .}

<p>for some functions 
  
    
      
        f
      
    
    {\displaystyle f}
  
 and 
  
    
      
        g
      
    
    {\displaystyle g}
  
.
</p><p>The Choquet integral does satisfy the following properties.
</p>
<h3>Monotonicity</h3>
<p>If 
  
    
      
        f
        ≤
        g
      
    
    {\displaystyle f\leq g}
  
 then
</p>

(
        C
        )
        ∫
        f
        
        d
        ν
        ≤
        (
        C
        )
        ∫
        g
        
        d
        ν
      
    
    {\displaystyle (C)\int f\,d\nu \leq (C)\int g\,d\nu }

<h3>Positive homogeneity</h3>
<p>For all 
  
    
      
        λ
        ≥
        0
      
    
    {\displaystyle \lambda \geq 0}
  
 it holds that
</p>

(
        C
        )
        ∫
        λ
        f
        
        d
        ν
        =
        λ
        (
        C
        )
        ∫
        f
        
        d
        ν
        ,
      
    
    {\displaystyle (C)\int \lambda f\,d\nu =\lambda (C)\int f\,d\nu ,}

<h3>Comonotone additivity</h3>
<p>If 
  
    
      
        f
        ,
        g
        :
        S
        →
        
          R
        
      
    
    {\displaystyle f,g:S\rightarrow \mathbb {R} }
  
 are comonotone functions, that is, if for all 
  
    
      
        s
        ,
        
          s
          ′
        
        ∈
        S
      
    
    {\displaystyle s,s'\in S}
  
 it holds that
</p>

(
        f
        (
        s
        )
        −
        f
        (
        
          s
          ′
        
        )
        )
        (
        g
        (
        s
        )
        −
        g
        (
        
          s
          ′
        
        )
        )
        ≥
        0
      
    
    {\displaystyle (f(s)-f(s'))(g(s)-g(s'))\geq 0}
  
.
which can be thought of as 
  
    
      
        f
      
    
    {\displaystyle f}
  
 and 
  
    
      
        g
      
    
    {\displaystyle g}
  
 rising and falling together
<p>then
</p>

(
        C
        )
        ∫
        
        f
        d
        ν
        +
        (
        C
        )
        ∫
        g
        
        d
        ν
        =
        (
        C
        )
        ∫
        (
        f
        +
        g
        )
        
        d
        ν
        .
      
    
    {\displaystyle (C)\int \,fd\nu +(C)\int g\,d\nu =(C)\int (f+g)\,d\nu .}

<h3>Subadditivity</h3>
<p>If 
  
    
      
        ν
      
    
    {\displaystyle \nu }
  
 is 2-alternating, then
</p>

(
        C
        )
        ∫
        
        f
        d
        ν
        +
        (
        C
        )
        ∫
        g
        
        d
        ν
        ≥
        (
        C
        )
        ∫
        (
        f
        +
        g
        )
        
        d
        ν
        .
      
    
    {\displaystyle (C)\int \,fd\nu +(C)\int g\,d\nu \geq (C)\int (f+g)\,d\nu .}

<h3>Superadditivity</h3>
<p>If 
  
    
      
        ν
      
    
    {\displaystyle \nu }
  
 is 2-monotone, then
</p>

(
        C
        )
        ∫
        
        f
        d
        ν
        +
        (
        C
        )
        ∫
        g
        
        d
        ν
        ≤
        (
        C
        )
        ∫
        (
        f
        +
        g
        )
        
        d
        ν
        .
      
    
    {\displaystyle (C)\int \,fd\nu +(C)\int g\,d\nu \leq (C)\int (f+g)\,d\nu .}

<h2 id="alternative-representation">Alternative representation</h2>
<p>Let 
  
    
      
        G
      
    
    {\displaystyle G}
  
 denote a <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> such that 
  
    
      
        
          G
          
            −
            1
          
        
      
    
    {\displaystyle G^{-1}}
  
 is 
  
    
      
        d
        H
      
    
    {\displaystyle dH}
  
 integrable. Then this following formula is often referred to as Choquet Integral:
</p>

∫
          
            −
            ∞
          
          
            ∞
          
        
        
          G
          
            −
            1
          
        
        (
        α
        )
        d
        H
        (
        α
        )
        =
        −
        
          ∫
          
            −
            ∞
          
          
            a
          
        
        H
        (
        G
        (
        x
        )
        )
        d
        x
        +
        
          ∫
          
            a
          
          
            ∞
          
        
        
          
            
              H
              ^
            
          
        
        (
        1
        −
        G
        (
        x
        )
        )
        d
        x
        ,
      
    
    {\displaystyle \int _{-\infty }^{\infty }G^{-1}(\alpha )dH(\alpha )=-\int _{-\infty }^{a}H(G(x))dx+\int _{a}^{\infty }{\hat {H}}(1-G(x))dx,}

<p>where 
  
    
      
        
          
            
              H
              ^
            
          
        
        (
        x
        )
        =
        H
        (
        1
        )
        −
        H
        (
        1
        −
        x
        )
      
    
    {\displaystyle {\hat {H}}(x)=H(1)-H(1-x)}
  
.
</p>
<ul><li>choose 
  
    
      
        H
        (
        x
        )
        :=
        x
      
    
    {\displaystyle H(x):=x}
  
 to get 
  
    
      
        
          ∫
          
            0
          
          
            1
          
        
        
          G
          
            −
            1
          
        
        (
        x
        )
        d
        x
        =
        E
        [
        X
        ]
      
    
    {\displaystyle \int _{0}^{1}G^{-1}(x)dx=E[X]}
  
,</li>
<li>choose 
  
    
      
        H
        (
        x
        )
        :=
        
          1
          
            [
            α
            ,
            x
            ]
          
        
      
    
    {\displaystyle H(x):=1_{[\alpha ,x]}}
  
 to get 
  
    
      
        
          ∫
          
            0
          
          
            1
          
        
        
          G
          
            −
            1
          
        
        (
        x
        )
        d
        H
        (
        x
        )
        =
        
          G
          
            −
            1
          
        
        (
        α
        )
      
    
    {\displaystyle \int _{0}^{1}G^{-1}(x)dH(x)=G^{-1}(\alpha )}
  
</li></ul>
<h2 id="applications">Applications</h2>
<p>The Choquet integral was applied in image processing, video processing and computer vision. In behavioral decision theory, <a href="/facts/Amos_Tversky/DfvE3URJ">Amos Tversky</a> and <a href="/facts/Daniel_Kahneman/YnI99yAy">Daniel Kahneman</a> use the Choquet integral and related methods in their formulation of cumulative prospect theory.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Nonlinear_expectation/mqG3txCP">Nonlinear expectation</a></li>
<li><a href="/facts/Superadditivity/hc4kIlu3">Superadditivity</a></li>
<li><a href="/facts/Subadditivity/jfRLRWfX">Subadditivity</a></li></ul>
<h2 id="notes">Notes</h2>

<h2 id="further-reading">Further reading</h2>
<ul><li>Gilboa, I.; <a href="/facts/David_Schmeidler/StfLuvU0">Schmeidler, D.</a> (1994). "Additive Representations of Non-Additive Measures and the Choquet Integral". <i>Annals of Operations Research</i>. 52: 43–65. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF02032160">10.1007/BF02032160</a>.</li>
<li>Even, Y.; Lehrer, E. (2014). "Decomposition-integral: unifying Choquet and the concave integrals". <i><a href="/facts/Economic_Theory_(journal)/AeqDzFbp">Economic Theory</a></i>. 56 (1): 33–58. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs00199-013-0780-0">10.1007/s00199-013-0780-0</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3190759">3190759</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:1639979">1639979</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Choquet, G. (1953). "Theory of capacities". Annales de l'Institut Fourier. 5: 131–295. doi:10.5802/aif.53. <a href="https://doi.org/10.5802%2Faif.53" target="_blank">https://doi.org/10.5802%2Faif.53</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Denneberg, D. (1994). Non-additive measure and Integral. Kluwer Academic. ISBN 0-7923-2840-X. <a href="0-7923-2840-X" target="_blank">0-7923-2840-X</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Grabisch, M. (1996). "The application of fuzzy integrals in multicriteria decision making". European Journal of Operational Research. 89 (3): 445–456. doi:10.1016/0377-2217(95)00176-X. <a href="/wiki/European_Journal_of_Operational_Research" target="_blank">/wiki/European_Journal_of_Operational_Research</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Chateauneuf, A.; Cohen, M. D. (2010). "Cardinal Extensions of the EU Model Based on the Choquet Integral". In Bouyssou, Denis; Dubois, Didier; Pirlot, Marc; Prade, Henri (eds.). Decision-making Process: Concepts and Methods. pp. 401–433. doi:10.1002/9780470611876.ch10. ISBN 9780470611876. <a href="9780470611876" target="_blank">9780470611876</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Sriboonchita, S.; Wong, W. K.; Dhompongsa, S.; Nguyen, H. T. (2010). Stochastic dominance and applications to finance, risk and economics. CRC Press. ISBN 978-1-4200-8266-1. <a href="978-1-4200-8266-1" target="_blank">978-1-4200-8266-1</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Lust, Thibaut & Rolland, Antoine. (2014). 2-additive Choquet Optimal Solutions in Multiobjective Optimization Problems. Communications in Computer and Information Science. 442. 256-265. 10.1007/978-3-319-08795-5_27.
 <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Tversky, A.; Kahneman, D. (1992). "Advances in Prospect Theory: Cumulative Representation of Uncertainty". Journal of Risk and Uncertainty. 5 (4): 297–323. doi:10.1007/bf00122574. S2CID 8456150. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
</ol>

Choquet integral open-in-new

Choquet integral